At times I insert a non-breaking space (on my Mac) instead of characters in a Last Name field when filling out forms. Is there a way to do this using the default or some other keyboard on my Nexus 4 (4.2.2)?
You might want to try with Hacker's Keyboard. They offer "compose key" functionality, which can be used for non-breaking space and more:
See the "circle key" to the right of the space bar? If you long-press it, according to the FAQ, this activates "compose key mode". They seem to follow the "standard rules" here, so "2x space" should give you a non-breaking space. A list of supported compose sequences can be found in the code.
A workaround is to copy the non-breaking space from somewhere (e.g. the Wikipedia page has it in the first line between the quotes), or from between the brackets here: